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They are written in terms of a Gaussian measure on the dual of $\\mathrm{U}(N)$ introduced recently by the author and M. Ma\\\"ida \\cite{LM3}. From these formulas, we prove a quantitative result on the convergence of the expectation and variance as $N$ tends to infinity, refining a result of the author and A. Dahlqvist \\cite{DL}. 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